State the principle of isolated zeros for an analytic function on a domain in $\mathbf{C}$.

Suppose $f$ is an analytic function on $\mathbf{C} \backslash\{0\}$, which is real-valued at the points $1 / n$, for $n=1,2, \ldots$, and does not have an essential singularity at the origin. Prove that $f(z)=\overline{f(\bar{z})}$ for all $z \in \mathbf{C} \backslash\{0\}$.